相关论文: Decay of the Kolmogorov $N$-width for wave problem…

The paper studies the rate of convergence of the weak Euler approximation for It\^{o} diffusion and jump processes with H\"{o}lder-continuous generators. It covers a number of stochastic processes including the nondegenerate diffusion…

Traditional linear subspace reduced order models (LS-ROMs) are able to accelerate physical simulations, in which the intrinsic solution space falls into a subspace with a small dimension, i.e., the solution space has a small Kolmogorov…

We derive a $L^1_x (\mathbb R^d)-L^{\infty}_x ( \mathbb R^d)$ decay estimate of order $\mathcal O \left( t^{-d/2}\right)$ for the linear propagators $$\exp \left( {\pm it \sqrt{ |D|\left(1+ \beta |D|^2\right) \tanh |D | } }\right), \qquad…

Given a domain $D$ in $\mathbb{C}^n$ and $K$ a compact subset of $D$, the set $\mathcal{A}_K^D$ of all restrictions of functions holomorphic on $D$ the modulus of which is bounded by $1$ is a compact subset of the Banach space $C(K)$ of…

In this work, we show that for the martingale problem for a class of degenerate diffusions with bounded continuous drift and diffusion coefficients, the small noise limit of non-degenerate approximations leads to a unique Feller limit. The…

We consider a degenerate infinite dimensional stochastic Hamiltonian system with multiplicative noise and establish the essential m-dissipativity on $L^2(\mu^{\Phi})$ of the corresponding Kolmogorov (backwards) operator. Here, $\Phi$ is the…

We propose local space-time approximation spaces for parabolic problems that are optimal in the sense of Kolmogorov and may be employed in multiscale and domain decomposition methods. The diffusion coefficient can be arbitrarily rough in…

We study the optimality of the minimax risk of truncated series estimators for symmetric convex polytopes. We show that the optimal truncated series estimator is within $O(\log m)$ factor of the optimal if the polytope is defined by $m$…

Numerical multiscale methods usually rely on some coupling between a macroscopic and a microscopic model. The macroscopic model is incomplete as effective quantities, such as the homogenized material coefficients or fluxes, are missing in…

We study some dyadic models for incompressible magnetohydrodynamics and Navier-Stokes equation. The existence of fixed point and stability of the fixed point are established. The scaling law of Kolmogorov's dissipation wavenumber arises…

The Gaussian width is a fundamental quantity in probability, statistics and geometry, known to underlie the intrinsic difficulty of estimation and hypothesis testing. In this work, we show how the Gaussian width, when localized to any given…

We derive quantitative error bounds for deep neural networks (DNNs) approximating option prices on a $d$-dimensional risky asset as functions of the underlying model parameters, payoff parameters and initial conditions. We cover a general…

We have made a new evaluation of the Lambda decay width in nuclear matter within the Propagator Method. Through the Local Density Approximation it is possible to obtain results in finite nuclei. We have also studied the dependence of the…

In this paper, we are concerned with the 2D damped wave-type magnetohydrodynamic system (abbreviated as MHD-wave system). The purpose of this paper is to study the large time behavior of solutions to the MHD-wave system, espesically to…

We construct a three-dimensional, fully relativistic numerical model of a universe filled with an inhomogeneous pressureless fluid, starting from initial data that represent a perturbation of the Einstein-de Sitter model. We then measure…

Second-order optimization methods, such as cubic regularized Newton methods, are known for their rapid convergence rates; nevertheless, they become impractical in high-dimensional problems due to their substantial memory requirements and…

We study a cutting-plane method for semidefinite optimization problems (SDOs), and supply a proof of the method's convergence, under a boundedness assumption. By relating the method's rate of convergence to an initial outer approximation's…

Accurate approximation of scalar-valued functions from sample points is a key task in computational science. Recently, machine learning with Deep Neural Networks (DNNs) has emerged as a promising tool for scientific computing, with…

In the problem of semialgebraic range searching, we are to preprocess a set of points in $\mathbb{R}^D$ such that the subset of points inside a semialgebraic region described by $O(1)$ polynomial inequalities of degree $\Delta$ can be found…

Many popular learning algorithms (E.g. Regression, Fourier-Transform based algorithms, Kernel SVM and Kernel ridge regression) operate by reducing the problem to a convex optimization problem over a vector space of functions. These methods…